Theorem breq12 4007

Description: Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)

Assertion

Ref	Expression
breq12	|- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Proof of Theorem breq12

Step	Hyp	Ref	Expression
1		breq1 4005	. 2 |- ( A = B -> ( A R C <-> B R C ) )
2		breq2 4006	. 2 |- ( C = D -> ( B R C <-> B R D ) )
3	1, 2	sylan9bb 462	1 |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   class class class wbr 4002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003

This theorem is referenced by:  breq12i  4011  breq12d  4015  breqan12d  4018  posng  4697  isopolem  5820  poxp  6230  rbropapd  6240  ecopover  6630  ecopoverg  6633  ltdcnq  7393  recexpr  7634  ltresr  7835  reapval  8529  ltxr  9771  xrltnr  9775  xrltnsym  9789  xrlttr  9791  xrltso  9792  xrlttri3  9793  xposdif  9878  exmidsbthrlem  14630